As far as AKC Champions, that is conformation championship, not temperament or working ability.
Conformation is all about how the dog is structurally put together i.e do they have a decent coat, all their limbs measure out and do they have both testicles and all of their teeth, etc.
Many show champions have dominant or recessive traits they continue passing onto offspring but it is up to each individual breeder to test for certain congenital conditions inherent and endemic to that particular breed. For example, St. Bernards, Newfoundlands, Rottweilers, Chesapeake Bay Retrievers, Golden Retrievers, German Shepherd Dogs, and Labs have the highest incidence of hip dysplasia. Miniature and toy Poodles, Maltese,Jack Russell Terriers, Yorkies, Poms and others have the highest incident of patellar luxation. There are other breed specific diseases and structural issues like degenerative myelopathy and heart problems as well as eye issues. These are just the most commonly discussed that we are familiar with.
http://www.hsvma.org/assets/pdfs/guide-to-congenital-and-heritable-disorders.pdf
A quality breeder who cares about their dogs and the dogs they bring into this world will test their breeding dogs for the problems most prevalent in their breed of choice. Quality German Shepherds that will have the best chance of being healthy will come from generations of OFA certified hips and DM-clear (Degenerative Myelopathy), for example. But that doesn't mean they can't develop these all the same. It just cuts down the chances.
The idea that mixed-breed dogs are healthier is a myth.
http://www.petmd.com/blogs/thedaily.../dispelling_the_mixed_breed_health_myth-12067 What offthegrid said is true, true true.
Now, we have Champion Dogs who are champions because of how they look, not their working ability as pertains to the breed standard. For example, Labrador Retrievers are shown in the Sporting Group because they are sporting dogs who retrieve things like duck and pheasant. However, the AKC breed standard that judges go by when assessing a potential champion Labrador Retriever has a huge list of appearance traits and only the following for temperament: "Temperament: True Labrador Retriever temperament is as much a hallmark of the breed as the "otter" tail. The ideal disposition is one of a kindly, outgoing, tractable nature; eager to please and non-aggressive towards man or animal. The Labrador has much that appeals to people; his gentle ways, intelligence and adaptability make him an ideal dog. Aggressiveness towards humans or other animals, or any evidence of shyness in an adult should be severely penalized."
https://www.akc.org/breeds/labrador_retriever/breed_standard.cfm
Nothing about hunting. Just basically be a nice dog and don't bite anyone. This has nothing to do with their ability to retrieve during a hunt. This is why there are Field Labs and Show Labs. Show Labs are champions in conformation. Field Labs can be champions too - not in AKC, but in sporting dog trials.
Same with German Shepherds. There are American Lines, German Show Lines, German Working Lines, etc. American Lines would never win in Germany and German Show Lines would never win in AKC shows. Working line dogs would extremely rarely win in the show ring because it's very tough to get beauty and brains all in one and the working lines are definitely the brains, but let me say again - sometimes will come a dog who has it all, but the AKC isn't going to find that dog and most people won't just happen to find that dog, but people who are working their dogs in real world scenarios and/or titling their dogs in certain competitions are the ones who will end up judging whether a dog "has it all". This is why, when choosing a dog, titles and/or work done does matter - it's the only way a dog is truly tested for their working ability in the area that breed is intended for. The AKC doesn't give one hoot about a dogs genetic predisposition, either medically or temperamentally. As long as the dog looks good and doesn't limp around the show ring or try to bite the judge. The AKC is a business. Having AKC papers doesn't matter. Having working (not show) titles does.
I have a German Shepherd that I chose very carefully and I didn't use a pedigree of champions. I used a pedigree of working dogs, with titles in Schutzhund, working K9s, Search and Rescue and Service dogs. I know the temperament of every dog in that pedigree going back several generations and the dogs out of those dogs. Well, the breeder, who also tests and titles all of her dogs, knew all of that - she just had to fill me in and pick a puppy for me out of her litter, because she knew those puppies the best during the eight weeks they had been alive and out of the puppy tests she gave them. I also know that I have a dog very unlikely to suffer from the congenital hip problems the breed is so prone to and I very nearly could have bet a lot of money on how my pup was going to turn out when he grew up. He is one now and exactly what we knew he would be. Knowing all of that was very helpful when choosing a dog for the specific task I chose this puppy for (Search work).
Adopting shelter puppies doesn't have to be a total dice roll. There are puppy temperament tests that can be done, depending on the age of the puppies, and a canine professional can definitely assess how a puppy is most likely going to turn out, independent, or insecure - which drives the puppy has, like prey drive, hunt drive, defense, are all going to be important to know so you can choose the puppy most likely to fit into your family and life situation. There are people who can help you choose your next puppy or dog based on their assessment of the animal. They can help you find the puppy who will grow up least likely to chase your cats or kill your chickens or defend your home by barking or protection.
To sum it up, having a line of champions doesn't mean you will have a dog with the appearance of a conformation champion but you will be more likely to. It will not mean you have a necessarily healthier dog or smarter more biddable dog, because looks aren't everything and the AKC means very little.
Having a line of working dogs may or may not mean that your dog will excel at the work the breed is meant to do, depending upon the handler, but you have a much greater chance of knowing what you will get. All of you with working LGDs know what I'm talking about here. I want my LGD from working LGDs and the AKC has nothing to do with their ability to do their jobs. But I can say right now that I will prefer a breeder who does certify hips because the LGDs are all large breed dogs prone to these problems.
Adopting a puppy may feel like an unknown, but it really isn't if you can find a qualified professional to help you assess the puppy and the same goes with shelter dogs. There are a lot of great dogs who die every day who would have been awesome pets and even working dogs. Why should anyone continue to breed unstable dogs, physically and temperamentally untested, including mixed and purebreds and add to the 2.7 million dogs who die in shelters each year, one dog killed about every 11 seconds? I don't have a good reason.